Let $E$ is a norm space. $\varnothing \neq A \subset E$ is open,convex set. $M$ is subspace of $E$. Suppose $A \cap M = \varnothing$. Then there exist a closed hyperplane $H$ such that $M \subset H$ and $H \cap M= \varnothing$
I see Hahn-Banach first geometric form: "Let $A \subset E$ and $B \subset E$ be nonempty convex subsets such that $A \cap B= \varnothing$. Assume that one of them is open. Then exist a closed hyperplane that separates $A$ and $B$".
Is it used to prove above proposition?